Question

what is the following simplified product? assume \\(x \ge 0\\).

\\(\left( \sqrt{6x^2} + 4\sqrt{8x^3} \
ight) \left( \sqrt{9x} - x\sqrt{5x^5} \
ight)\\)

• \\(3x\sqrt{6x} + x^4\sqrt{30x} + 24x^2\sqrt{2x} + 8x^5\sqrt{10x}\\)
• \\(3x\sqrt{6x} + x^4\sqrt{30x} + 24x^2\sqrt{2} + 8x^5\sqrt{10}\\)
• \\(3x\sqrt{6x} - x^4\sqrt{30x} + 24x^2\sqrt{2} - 8x^5\sqrt{10}\\)
• \\(3x\sqrt{6x} - x^4\sqrt{30x} + 24x^2\sqrt{2x} - 8x^5\sqrt{10x}\\)

Explanation:

Simplify each radical term

Using the Radical Simplification knowledge point

$$
LATEXBLOCK0
$$

Rewrite the product with simplified terms

Using the Multiplying Binomial Radicals knowledge point

$$ (x\sqrt{6} + 8x\sqrt{2x})(3\sqrt{x} - x^3\sqrt{5x}) $$

Expand the product using FOIL

Using the Simplifying Radical Products knowledge point

$$
LATEXBLOCK1
$$

Combine the expanded terms

Using the Simplifying Radical Products knowledge point

$$ 3x\sqrt{6x} - x^4\sqrt{30x} + 24x^2\sqrt{2} - 8x^5\sqrt{10} $$

Answer:

• (A) \(3x\sqrt{6x} + x^4\sqrt{30x} + 24x^2\sqrt{2x} + 8x^5\sqrt{10x}\)
• (B) \(3x\sqrt{6x} + x^4\sqrt{30x} + 24x^2\sqrt{2} + 8x^5\sqrt{10}\)
• (C) \(3x\sqrt{6x} - x^4\sqrt{30x} + 24x^2\sqrt{2} - 8x^5\sqrt{10}\) (Correct answer)
• (D) \(3x\sqrt{6x} - x^4\sqrt{30x} + 24x^2\sqrt{2x} - 8x^5\sqrt{10x}\)